Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations ∫_Sf(xyt)dµ(t)+∫_Sf(xσ(y)t)dµ(t)= 2f(x)f(y), x,y ∈ S; ∫_Sf(xσ(y)t)dµ(t)-∫_Sf(xyt)dµ(t)= 2f(x)f(y), x,y ∈ S; where S is a semigroup, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ _zi)_{I ∈ I}, such that for all i ∈ I, z_iis in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.